At every point this passage sounds to me like a rebuttal against the teachings of the Pharisees and scribes. Jesus says: You have heard that it was said, "Do not commit adultery." But let me clue you in on something. You are not as safely righteous as you think you are. First off, you have broken this commandment if you even look at a woman with lustful intent. Secondly, you have broken this commandment if you either divorce your wife (because now you're going to make her commit adultery) or if you marry a divorced woman. So don't go checking off the seventh commandment from your "to do" list as if it's something you can easily accomplish. You are more of a law-breaker than you think you are.
The Pharisees and scribes believed they were committed to righteousness. But Jesus says that real commitment to righteousness means tearing out the eye that can't stop looking lustfully, or cutting off the hand that might touch what is forbidden. At this point I guess I'm supposed to do what most Bible teachers do, and go on and on reassuring you that Jesus didn't literally mean that you should mutilate yourself. Of course I think that's true, mainly because Jesus teaches elsewhere that it's the evil inside your heart that causes us to sin, not a body part strictly speaking.
But let's set aside panic about this passage for a moment and take it seriously. You have to admit that Jesus' logic is flawless. Suppose that your eye or hand could cause you to stumble enough to send you to hell. Which is better, to keep the body part in this life and have your whole body perish in the next, or to lose the body part in this life and save your whole body in the next life? Well, duh, right? Now what do the Pharisees think of eyeless, handless, and generally mutilated people? They view them as damaged goods who should exist on the outskirts of their holy community. Perhaps Jesus is pointing out the physically mutilated person may be the truly whole person, while the apparently whole person may be the one who is destined to perish.
I am told that the Pharisees did not consider divorce a huge deal since Moses mentioned divorce in Deuteronomy 24:1-4. (Jesus later says that Moses only taught this "because of the hardness of your hearts" but in God's ideal world there would be no divorce at all. But that's for another Bible study.) They would have considered Jesus' teaching on divorce to be extreme, and no doubt Jesus is aiming to shock them. Now you may be wondering why divorcing one's wife automatically makes her commit adultery. She could just stay single, couldn't she? Not so. In those days a woman was completely economically dependent upon her husband, and so if her husband divorced her, she would be forced to find another man who'd be willing to marry her just so she could survive. She would be forced to commit adultery.
See, Jesus isn't one of these legalists who would agree with a man who says, "Well, I didn't force her to do anything. Sure, I divorced her, but she's the one who made the choice to get remarried and commit adultery. That was her choice not mine." But Jesus recognizes that the choice between getting sinfully remarried and starving to death is no choice at all. That is why he lays the blame at the feet of the woman's husband who put her in that position in the first place. He takes people's circumstances into account instead of just applying the Law in an impersonal, cookie-cutter fashion. Interesting, no?